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Full text of "Evaluation of cross-correlation methods in the utilization of antenna systems"

L I B HAHY OF THE U N IVER_SITY Of 1LLI NOIS €21.365 Tje€55te no. 40-^9 cop. 2 Digitized by the Internet Archive in 2013 http://archive.org/details/evaluationofcros49macp ANTENNA LABORATORY Technical Report No. 49 EVALUATION OF CROSS-CORRELATION METHODS IN THE UTILIZATION OF ANTENNA SYSTEMS by Robert H, MacPhie 25 January 1961 Contract AF33(616)-6079 Project No. 9-(13-6278) Task 40572 Sponsored by: WRIGHT AIR DEVELOPMENT CENTER Electrical Engineering Research Laboratory Engineering Experiment Station University of Illinois Urbana, Illinois ENGINEERING LIBKARY ACKNOWLEDGMENT The author is very grateful for the many helpful suggestions from Professor Deschamps. The discussions with Professor Lo^ Mr. Craig Allen and other members of the Antenna Laboratory were also of considerable' help, ABSTRACT The theory of the cross-correlation of signals of two or more antennas to produce antenna product patterns is reviewed and presented in a generalized form. The problem of distortion of amplitude modulation by the multiplicative processes is considered and a system to remove this distortion by filtering is proposed. An analysis is made of the correlation system output when in the presence of a primary desired signal there is a secondary interfering signal incident on the antenna system from a different direction. It is shown that the rejection of this interference does not depend entirely on the product pattern itself. Indeed^ if the interfering signals amplitude is relatively low, the interference in the system output is determined mainly by a low-level interference pattern which is essentially the sum of the factor patterns of the individual antennas rather than their product. Some ostensibly good product patterns have associated with them very poor low-level interference patterns. The analysis considers the generalization to more than one inter- fering signal and more than one pattern multiplication as well as the effect of time averaging (filtering) which is sometimes possible, for example, in the case of Radio Astronomy. Use is made of the negative sidelobes of the product pattern to suppress the integrated effect of average background noise. It is shown that for any arbitrary noise background with the signal known to be absent the time average of the total contribution of this noise in the system output can be reduced to zero. Then when the signal is transmitted its time averaged effect alone will be observed in the system output. It is assumed that the noise from any given direction is a quasi-stationary random process which is statistically independent of the noise from any other direction. A discussion of the noise limitation on the formation of arbitrary patterns from a two element array by means of multiplicative operations is presented and it is shown that any background noise will generally cause a shift in the apparent direction of arrival of the signal as 'seen" by the two element product pattern. The formation of an arbitrary pattern from this basic two element pattern with its error due to the shifting action of the noise does nothing to reduce this error. Rather^ it only serves to emphasize the apparent direction of arrival of the signal and if the noise is relatively large the error between the apparent direction and the true direction can be considerable. CONTENTS Page 1, I rr reduction 1 2, The Multiplication of Antenna Patterns bv Signal Cross-Correlation 3 2„1 Two Antenna Case 3 2„2 Demodulation and Multiplicative Distortion 5 2„3 N Antenna Case and Removal of Amplitude Modulation Distcr- ^ion by filtering 6 3, Analysis of the Effect of More Than One Plane Wave Incident en the Antenna System 13 3 1 0;+put of An Additive System with Two Plane Waves Incident 13 3„2 Output of a Two Antenna Multiplicative System with Two : nndpp' Plane Waves 14 3 3 Output of a Two Antenna Multiplicative System with R Interfering Signals 22 3.4 '•.■*" erf erence in the Case of More Than One Multiplication 24 4c Suppr-? -" <~>n ft T he Effect of Average Background Noise by Use of the Negative Sidelobes of the Product Pattern 28 5„ The NToise Lim ition on the Formation of Arbitrary Patterns from a Two Element Interferometer by Means of Multiplicative Operations 36 6, Cone 1 us i on' 42 Reference? 44 ILLUSTRATIONS Figure Number Page 1. Mul tipli ca T j en of Two Antenna Patterns 4 2. Frequency Spectra of Signals at Successive Stages of the Process of Removing Undesirable Modulation on the Signals to be Cross-Correlated. 8 3. System for Removing Modulation from All but the Last Product Signal for the Case of Multiplication of N Antenna Patterns 10 4. Comparison of Product Pattern and Low-Level Interference Pattern for the Compound Interferometer 19 5. System for the Suppression of the Average Value of the Output of a Compound Interferometer Due to an Arbitrary Distribution of Background Noise 34 6„ Phaser Representation of the Signal Plus Noise Output from a Two Element Multiplicative Array. 40 7. <a) Interferometer Pattern of Signal Alone, (b) Interferometer Pattern of Noise Alone, (c) Interferometer Pattern of Signal Plus Noise^ (d) Synthesized Pattern of Signal Plus Noise 42 INTRODUCTION For some time cross-correlation techniques have been used to obtain receiving antenna patterns which are effectively the product of the patterns of two or more different antennas. Practical examples of such systems are the Mill's 1 2 Cross and the Compound Interferometer both used in Radio Astronomy. In the former, two long, mutually perpendicular antennas of lengths L and L in the form of a cross, have their signals cross-correlated to produce a power pattern, with a pencil beam, roughly equal to the voltage pattern of an antenna whose area is L L , The Compound Interferometer places a uniformly weighted aperture beside a simple interferometer and the cross-correlated output combines the sharpness of the interferometer pattern with the directivity of the aperture. If both are of length $ (hence the system is of length 2i ) , the product power pattern is equal to the voltage pattern of a uniform aperture of length 4i. In this report the product pattern for the general case of two antennas separated by a distance i and with complex patterns is presented Then the further generalization to the multiplication of N antenna patterns is made, and a system to accomplish this is proposed. In this connection the problem of demodulation is considered. The multiplicative operations cause distortion of the modulating signal and it is shown that this multiplicative distortion can be eliminated (in theory) by filtering. The problem of rejecting interfering signals from arbitrary directions other than that of the desired signal is more complicated than in the case of additive arrays. It will be shown that the product pattern alone does not describe fully the rejection of interference. Another pattern, which in general depends on the relative directions of the primary and interfering signals, has to be introduced. In the case of a single cross-correlation the pattern is essentially the sum of the factor patterns of the two antennas rather than their product It will be shown that some good product patterns have associated with them rather poor low-level interference patterns „ More generally, if the signals of N antennas are cross-correlated, in the system output there are N-l distinct interference patterns all of which must be analysed when interference rejection is required. The product pattern, unlike the power pattern of additive antenna systems, has negative sidelobes and in Section 4 it is shown that, they can be used to eliminate the effect of background noise in the time-averaged output of the correlation system. The proposed scheme will work only when it is known a priori when the signal is present and when it is absent,, In addition the noise from any direction must be quasi-stationary and statistically inde- pendent of noise coming from any other direction„ Finally an analysis of the noise limitation on the formation of arbitrary patterns from a two element array by means of multiplicative operations is presented. It is shown that errors introduced by the noise into the two element product pattern cannot be reduced in the formation from i t of patterns of ostensibly much greater resolution. 2. THE MULTIPLICATION OF ANTENNA PATTERNS BY SIGNAL CROSS -CORRELATION 2,1 Two Antenna Case A diagram of the multiplication system for two antennas is shown in Figure 1. The signal from antenna A is A(9,t) = Re A(0)e Jwt volts (1) and that from antenna B is B(6,t) = Re B(e)e" jf3isin6 e Jwt volts (2) = Re B (Q) e JCOt volts (2a) where 9 = angle which the propagation vector of an incident plane wave makes with the normal to the line joining the two antennas, t = time in seconds, u> = angular frequency of the incident plane wave in radians per second, i = spacing between the antennas in meters, P = propagation constant in radians per meter, Re indicates 'the real part of". The pattern coefficients A(Q) and B (0) are in general complex. Note that the effect of the spacing i between the antennas is the phase factor e " K which is absorbed into B.(0)„ Now as is shown in Figure 1, the signal from antenna A enters a frequency shifter (a nonlinear device) together with a local signal at frequency oj . The output is filtered so as to pass only the difference frequency (oj = to - co ) LOCAL PUMP SIGNAL (J ANTENNA A 9 ANTENNA B ¥ W i > x A(0,<U) B^(0,W) FREQUENCY SHIFTER AND FILTER A(0,o/) It MIXER W. i Pp<0,W |f <) SYNCHRONOUS DETECTOR I R,(0,f) '' '• Multiplication of Two Antenna Patt erna signal which is simply the signal from A shifted in frequency to co . Then this signal and that from B enter a mixer which is a second nonlinear device whose output contains among other components a signal at frequency to . By means of a synchronous detector the amplitude of this signal can be obtained and it is given by p (e, i) = Re A(e) B*(e) o> ft where indicates the complex conjugate quantity. In trigonometric form, Equa- tion (3) can be written as p (e, i) = :A(e) B(e>l cos[pisine + a (e> - p(e)l (3a) where a(G) = Arg A(Q) and (3(9) = Arg B(Q). Since P (Q,4) is the product of two voltage patterns it varies as the power of the incoming waves and consequently any comparison with additive antenna systems patterns will be with the power patterns of those systems. We note incidentally from Equation (3) that if / = and A(Q) = B(Q), then the product pattern degenerates to an additive systems power pattern given by P o (0) = |A(e)^ 2 (4) I 2.2 Demodulation and Multiplicative Distortion I — . If the incoming plane wave is modulated, with amplitude modulation for example, then the output of the system will be P 2 (0, F(t),l) =[14 mF(t)] 2 Re A(e) B*(e) (5) = [1 + 2mF(t) + m F(t) ] Re A(e) B*(0) (5a) where m = percent modulation, (0 < m < 1), F(t) = modulating signal. 2 2 Hence there is square law distortion (m F(t) ) which can be significant for large m. This problem also occurs in additive systems which employ square law detection. Gray* has shown for the simple case of sinusoidal modulation that for less than 10$ square law distortion m must be less than 40$ which is not too severe a restriction. However, if N antenna patterns are multiplied together (as will be shown in the next section) the output is proportional to P N (G, F(t)) = [1 + m F(t)] N P (0) (6) As N increases the restriction rapidly becomes intolerable; for N = 6 and 10$ 2 2 distortion from only the m F(t) term in the binomial expansion, m must be kept less than 8$, In the following section a system for multiplying N antenna patterns which removes this multiplicative distortion is proposed. 2,3 N Antenna Case and Removal of Amplitude Modulation Distortion by Filtering If there were N antennas the spectrum of the signal at the terminals of each would be that of the carrier and its modulation while the amplitude and phase of the signal would be determined by the antenna's pattern and its loca- tion. Now since any one of the N antennas signals is modulated in the same way as any other, nothing would be lost if the modulation were removed from the carrier ol - • 1 1 but one of these signals. Not only will there be no loss of information bill L1 Will be shown that this actually eliminates the multiplicative distortion brouL'M aboul by the repeated cross-correlation of the antenna '•8 . 1 Applied Electronics, Wiley, pp. 748-740 Let the N antennas have terminal voltages given by A (0, t,F(t)), 1 A (0,t,F(t)) A (9,t,F(t)) where k N A, (G,t,F(t)) = [1 + mF(t)] Re A (0) e "J Cpi k sin0-ct) £ k k = [1 + mF(t)] Re A (0) e'^ 1 (7) k For simplicity we consider the case of colinear antennas whose patterns are functions of alone. The more general case of non-colinear antennas with "two-dimensional" patterns (Q,§) is of no greater difficulty as far as the multiplicative operations are concerned In Figure 2(a) are shown the spectra of A (0, t, F( t)) and of the local signal at frequency co . The separation of 1 the local signals frequency co from the carrier's at to is A co and is chosen soas to be slightly larger than the spectral width of one of the carrier's sidebands. Hence, the output of the first frequency shifter (FS) will contain the difference frequency at A co along with the modulation spectrum of the signal This difference frequency can be held constant by an automatic frequency control device (AFC). The low frequency output spectrum is shown in Figure 2(b). A high Q filter tuned to frequency A co will remove the unwanted modulation on the new "carrier" leaving a single frequency signal whose amplitude varies as the antenna pattern A. (0), (see Figure 2(c)). The frequency shifting operation 1 to a much lower frequency is made necessary by the difficulty of removing the modulation from an RF carrier by direct filtration. A second frequency shifter (FS) ? this time with a local signal at frequency U = co - 2 A co is used to return the unmodulated "carrier" at A co to the RF SIGNALS INTO FIRST FREQUENCY SHIFTER RADIO (a) FIRST PUMP FREQUENCY . ... n/ ,. Q . FREQUENCIES 4 ♦ A^.FM.e) nrv . J. cj ^^ (b) .1 LOW FREQ. OUTPUT FROM FIRST FREQ. SH ♦ A{(A£J,F(6)),e) = A{«J,F(a»,G) SHIFTER DOWN y<"~"^ /^N IN FREQUENCY TO AW FTER |A^(AW,e) /MODULATION SPECTRUM F(U)\ \REMOVED BY HIGH Q FILTER/ A6J (d) i SIGNALS INTO SECOND FREQ. SHIFTER | A^(A6),9) 1 At) 2ND PUMP FREQUENCY (e) T 2 2 Ad) SIGNALS OUT OF 2ND FREQ. SHIFTER A {| (6J-3A^J,e) SUPPRESSED CARRIER A.(6J-A6J,0) ■li ii SIGNALS INTO FIRST MIXER (f) CJ-3AO) Gi^ GO-ACJ O) t A < (g) W-3A6) OW MIXER 2 a., (6),F(6)),e CJ-ACJ 0) P 2 AlA^I' |P 2 i/ir\17ir\ A<0 2AOJ 3A6J (h) "P (ACJ.9) /MODULATION SPECTRUM F«J) V REMOVED BY HIGH Q FILTER; Figure 2. FREQUENCY, RAD./SEC. PROCESS CAN BE CONTINUED BY STARTING AT STAGE (d) ABOVE AND USING A. {CJJ{Q),Q) IN THE NEXT MIXER OPERATION BCy Spectra of Signals at Successive Stages of the Process of Bmovlng f.'ndosi rable Modulation on the Signals to be Cross-Correlated, range. The FS output will be a new carrier at frequency oo with sidebands at oo - A oo and oo + A co. At this point it is necessary to suppress the carrier (there is no technical difficulty in doing this), Then the two sidebands whose amplitudes are proportional to A (Q) are mixed with the modulated signal from 1 antenna two, A (0, t,F(t)) as is shown in Figure 2(f). The low frequency 2 difference output spectra are shown in Figure 2(g). Again an AFC device is used to hold this output constant in frequency. A filtertuned to A co passes the product signal given by P 2 (9,t, i 1 ,i 2 ) = Re A i (0) A i *(0) e j Aw t (8) and which has all its modulation removed. The process can be continued by returning this signal to RF as was done in stages 2(d) and 2(e) (see Figure 2). Then by mixing with A ,(0, t,F(t)) and filtering one obtains P 3 (0,t,i 1 ,i 2 ,i 3 ) = Re A i (0) A i (0) A jf (0) e -I *-i O j A oo t (9) which again has all the modulation removed. A diagram of the system is shown in Figure 3. The process is continued until the (N - 2)ND unmodulated pro- duct is formed, i.e., N-2 j A w t P N _ 1 (0,t,i i ,...i N _ 1 ) = Re A A (0) A *(0) 4 x 2q-l 2q A (0) e^ N-l if N is even and > 3, and (10) Vl^^l'-W =R N-l n a ( Q ) a *(0) 4 2q-l 2q j A oo t if N is odd and > 2 r (in the above formula i is zero), Then, instead of c bo •H CO O 3 T3 O 0, W 3 J2 3 •H ■o r ( a a •H ■^ bfl H e 3 •H 3 > <H e o 0) ad a h Ed U >H 0) a d u +J M ■ Fi >) 03 <M 3 ho •H 11 shifting this signal back up to RF, we mix it directly with the modulated signal from the last antenna A (0, t,F(t)), The modulated RF difference frequency is N if the mixer output signal at this frequency is fed into a synchronous detec- tor along with a reference signal also at co taken from the local source, then the detector output will be N 2 ^(0, F(t) £ 2? l 3 ,...l }i ) = [1 + m F(t)] Re U A < 0) A 2q-l (0) A, *(Q) 2q (11) if N is even and > 1 and P N (0, F(t), i 2 ,i 3 , ...,1 N ) = [1 + m F(t)] Re f N-l q 2i A i (e) A % <°> 4 X 2q-1 *2q A iN (8) if N is odd and > 2. The output has no multiplicative distortion and consequently m need not be limited to say, less than 40$ as with a square-law detector. The practical success of the system depends on the filtering efficiency which in turn depends on the modulation spectrum. The noise problem should be alleviated by the filtering but the extra multiplicative operations will clearly degrade the signal to noise ratio. Finally, mention should be made of amplification which probably will be necessary if N is large. The most convenient place for an amplifier 'would be immediately after the filter that removes the modulation from the pseudo-carrier at A co. A narrow band low noise parametric amplifier could be ised. It should be noted that the pattern P ( 0, F . __ . , i_, . . , ,i ) is proportional N (t) 2 ' N ;o the Nth power of the electric field strength of the incoming plane wave, tence the equivalent "field strength pattern" (voltage) is T> t Q V t +■ \ \ — ID (d T?f + \ e » 1 ,N l 1 CM 12 and the equivalent power pattern is 2 p N p (e,F(t)i 2 , ....i N ) = P N ( Q ,F(t),i 2 , ... i N ) N (is) Any comparison with conventional additive patterns should be made on this basis. In the rest of this report only the power pattern will be used and the superscript will be omitted. 13 3. ANALYSIS OF THE EFFECT OF MORE THAN ONE PLANE WAVE INCIDENT ON THE ANTENNA SYSTEM In the preceding analysis, only a single plane wave incident at the arbitrary angle 9 was considered. In this section it will be shown that when two plane waves are incident at angles 9 and Q , the output of the synchronous detector of the product system is in important respects different from the output of the conventional square law detector of additive systems. We let one of the plane waves of unit amplitude at an angle of incidence 9 be the desired signal. The other, of complex amplitude Se \ and angle Q , is either due to a discrete, remote, noise source or a secondary component of the desired signal which arises when there is more than one transmission path from the source to the receiver (multipath propagation). 3.1 Output of An Additive System wit h Two Plane Waves Incident In an additive system the received terminal voltage before detection is a linear superposition of the two signals C(t;l,9;Se , %) = C(t;l,9) + C(t;Se ,S ) = Re]c(9) + Se'^1 CCQ^ \ e jwt (14) The usual detector > a square law device, has a low frequency output proportional to . t , 2 2 2 .t Ca^iSeW^ = !c(9)^ + S C(9j + 2S Re C(0; C(Q) e" j5i (1! l ■ ' i i 1 )r alternatively 3 Welsby and Tucker have obtained some similar results in which they compare multiplicative systems with additive systems employing linear rather than square law detection. 14 I i ^l I 2 I I 2 2 1 I 2 , |cu,e;Se J , e 1 )| = |c(e)| + s |c(e 1 )| + 2s|c(e) c(e 1 )|cos l 1 -Ke)+4(e 1 ) (15a) where £{Q) = ARG C(9) . o If the pattern is normalized with the primary signal incident at = i the beam angle of the pattern, Equation (13a) can be written as c(se J , 1 e 1 '> = 1 + s 0(9^ + 2s|c(e i )|cos(l 1 -t(0°)+^(e i )^ (16) C(G 1 ) where c(9 ) = , The last two terms of Equation (16) represent interference and its rejection is seen to depend on c(0 ) alone. 3.2 Output of a Two Antenna Multiplicative System with Two Incident Plane Waves The output of the synchronous detector in the two antenna multiplicative case and with two incident plane waves is naturally more involved than that of the additive system. It is of the form P 2 a,9;Se j ^ 1 9 1 ;i) = ReJA(e)B i *(e) + S 2 Af e^B^^ + s(A(9)B i *(9 1 ) e ' l + A(0 l )B i *(9) e * (17) or alternatively (and with some rearranging) ,h P (l,e;SB , 9,;i) = A(6) B(Q) cos r : p/sinn-:a(0.'-(3<0> & 1 s a(Oj > B(e 1 )|cos lysine +a(e 1 )-p^e 1 )] A(9) B(9)|cofl p;sinG-ta(0)-|i(9)] ■ A(Q i | a(e 1 )-p(e) l I n6 a(6 I -p(6) B(9j ) bo) cos| pislnSj -i a(0)-p(e,)] i i no n(f)J |ii n ' (17a) 15 Now as in the additive case, if the pattern is normalized with the pri- mary signal incident at 6 = 0° Equation (15a) becomes P 9 (Se J , 1 Q t) = 1 + s 2 ia(9.)b(e i ) cos[ pisine Td(e )-p(e )] ~" c M[a(0° > ^(M o 5 TT " ~~ + S ', ia(6, ) cos(e i ^a(e 1 )-p(o°)) II cos(a(0°)-p(0°)) + ib(e,) i J- cos(pisin0 1 -^ 1 ^a(O°)-p(e i )) cos(a(0O)-(3(0°)) (18) J A(e 1 ) B(e 1 ) where 0.(0^ = ^^J . b 'V = j^ • Comparing Equations (16) and (18) we note that the basic difference is in the last terms which is the major interference term when S <* 1. In the additive case (Equation (16)) the rejection of interference depends on the voltage pattern c(0 ): the worst case occurs when £ = C(O)-C(0.,) and the interference 1 11 term is 2S c(0 )| , In the multiplicative case (Equation (18.)) the low-level interference is proportional, not to the product pattern itself but to the sum of the normalized factor patterns | a(0 )| and lb(0 )| each of which is multi- 1 i 1 plied by a cosine term. Consequently the rejection of this first order ( - s) interference by a multiplicative system cannot be determined by the product pattern at all. It will be convenient to define the pattern associated with S as the low-level interference pattern since it predominates when S is small and & is negligible Hence we let ,ar0 )| cos(£ -q(0 1 )-(3(O°)) * IbO,)! cos(p.fsin0 -S.'crtcft-pfe)) cos(a(0°) - (3(0°)) This expression is clearly equal to or less than ja(0 i )i -» jb(0 1 ) (19) but the 2cos[ a(0°)-p(0°)] exact nature of the pattern is quite complicated for this general case. Now 16 for any given system, i is fixed and the pattern is a function of the two variables G-, and £ „ Likewise for any given Q there will be an £ which ma ximizes the magnitude of L(Q ,§ ,i). To determine this we differentiate with respect to £ and equate the derivative to zero. 8 L<Q i ,£ v O 1 -laCe^l sin[^ 1 +a(e i )-p(0°)]+|b(e i )| sin(pisine i -S 1 +a(0 o )-p(6 1 )) cos(a(0°)-p(0°)) = (20) from which we obtain the relation la(0 )lsin(a(0 )-p(O°))-ib(0 )lsin (pfsin G.+a(O°)-p(0 )) i- e i ,i; |" ra(e 1 )!cos(a(e 1 )-p(o°)T+ib(e 1 )lcos(piiin e 1 +a(o°>-p(e 1 )) -p<e r (21) Now there are two values of £,(Q,i) (in the range [ 0, 27T]) that satisfy the above equation, namely § (6,i) and £ (0 ,i) + 77. Substituting these values of £ into the expression for L(0 , 1, -O gives the worst possible patterns L(0 £ ,i ) and L(0 |.+ 7T, i), If the patterns a(0 ) and b(0 ) are real, Equation (21) reduces to € j (0 I) ■ tan b(9 ) sin (P i sin ) a(0 ) + b(0 ) cos (P 1 sin Gj) (21a) and the expr< lor the worst patterns are '.'V V " a(9 1 )+b(9 1 ) cos(Pisin Oj) ! 2 + | b(Q ] ) si n (Pi sin ) . ' a^Oj) + b(0 ) cosfPisin 9, ) (22) cos (1 (9j , i )) 17 ue^ l I+ ir,i) a(8,)tb(eJ cos(Pisin ) I 2 + J b(0 n ) sin(Pi sin n K 2 1 _1_ 1_ L _ 1 1 J a(0 ) + b(0 ) cSsTPisin ) (22a) cosf£ (0 I) + TT) But cos (I (0 , I ) + 77) = -cos (| (0 i)) and hence L(e x , Ij + ir,i) = -L(0 r e p I) (23) Subsequently we will consider only L o (0 r i) = L(0 r e r i) (24) knowing that the alternative solution L(0 £, + 77 ? f ) is simply its negative. If i = there is a further reduction to L (0 ) = - [a(0 ) + b(0 )] o 1 2 1 1 (22b) Equation (22) can be put into a quite symmetric form by letting M(0 , I) = a(0 ) + b(0 ) cos(P I sin ) (25) and N(0 r i) = b(0 ) sin(P I sin ) (26) Then Equation (22) becomes W U -2 M(e x , i) 2 + n(0 i) 2 cos tan .1 N( 6p i> M(0 £) (27) 18 We can therefore conclude that some apparently good product patterns might have quite unacceptable low level interference patterns. As an example of this we consider the Compound Interferometer mentioned in the introduction. It consists of a uniformly weighted aperture of length i, adjacent to a simple interferometer also of length i. A diagram of the system is shown in Figure 4, The normalized factor patterns are sin P | sin 9 \ a(e 1 ) . _v 2 P i sin 0, b(9 1 ) = cos "'P i sin 9. Hence the product pattern is proportional to P I sin 0. sm p 2 ( 8i , D---PT P i sin A sin 9 n cos V cos (P I sin ) sin /'P £ sin fl p 2 ,e 1 - i "' -tn P i sin 9, sin 9, where x = — - — sin(4x) 4x ' (28) But the worsl l ->w level interference pattern is L (x) o 2 x 2 , , „ v 2 cos x cos 2x f (cos x sin 2x) sin x + cos x cos 2x , s i n x . / • cos x cos 2x tan ' - c:ds x sin 2x (28a) 19 O) C/5 V ( \ r \ V CT < I- O Q O cr Q. UJ O z. Ul cr ui Li- ce: UJ UJ > Z uj or 5 o . t= 4 ) 0) as a, 0) u c a> i« a> 0) CD > M o +j •c a a) 0) C w CD -u •o c 3 o o, s o o CD o o en <h •r-l h IS s O O 0) 3 CUD •H 20 The two patterns are plotted in Figure 4 as functions of x. It should be noted that this graph shows only the normalized patterns (each with its max- imum equal to unity). In any particular case, the relative magnitudes of the two patterns depend on the value of S. For example, suppose that S = 0.25 5 and corresponds to x = — 77. The output signal to interference ratio for 1 8 the worst case would be R(0 1 S) = — S P 2 (9 1 , 1) + 2S L o (0 p I) (.25) (.127) + 2(.25) .395 s = .25 ~ 577/8 00794 + .1974 (29) = 4.88 It can be seen that the low-level interference pattern dominates in this case because of its high sidelobes which are only 3 db down at x = +77, + 277 ( +377 etc. 2 Note that in some cases S P (0 , SL ) < and hence the worst case of interference would occur when 2S L (0,, 1) were negative also. If L (0,, i) o 1' o 1 ' is not negative as in the above example, we would select the alternate worst pattern -L (0, , 1 ) = L (0 , , £ _ + 77 t ) (see Equation (23)). The worst possible o 1 o 1' I signal to interference ratio for the system, without regard to sign, is therefore R <G., i, S f> 1 ' ' )l = S I P.O., 1)1 + 2S|L (0 i ) 2 1' o 1 ' (30) By way of comparison, the worst case for an additive system would be R (6 , S)| ■ 8 2 |c(9 1 )| 2 + 2810(8] » (31) 21 In the latter equation it is seen that high directivity of c(0 ) also means a large signal to interference ratio while the high "directivity'' of P (0 , I ) in the former equation can be negated by L (0 S.) to give a low net signal to interference ratio. It is important to realize^ however, that this situation obtains only when the interfering signal is at its worst possible phase with respect to the desired signal. Usually the interference phase £ is a time varying quantity with the probability of any one value of £ (t) in the interval [0, 27T] being the same as any other. Hence, the average of the low-level interference when taken over a suitably long period of time approaches zero, i.e.. 1 im T -r A(0 X ) Me)" cos (P I sin + |(t) + a(Q ) - (3(e)) ~cos (f3 £ "sin + cl(6) - (3(e) (32) P«9 , ) . cos ((3 i sin 0, - £(t) + a(0) - (3(0 )) 1 ft©; 1 r___ _ "cos Wl sin + o-(0) _r T(0)) dt = In practice this means that a low pass filter would be used to reduce the low-level interference signal in the system output. This filtering process is quite feasible in Radio Astronomy (hence the usefulness of Compound Inter- ferometer and indeed of the Mill's Cross), since only the steady state flux of energy from each point on the celestial sphere is required. For instan- taneous reception of modulated signals or for any application where a lengthy time averaging is not practicable, the low-level interference pattern must be considered. 22 3.3 Output of a Two Antenna Multiplicative System with R Interfering Signals If in addition to the desired signal of unit amplitude incident at the th angle 0, there are R secondary interfering signals^ the q of which has amplitude S e and direction , the output of the square law detector q q' of the additive system will be j£. j&, 6(1, S ie , i; ...; S R e , R ) R # -jb = |'C(6)| + 2Re fo'e) 2 S C (0 )e q q-1 q q r r : . Z SsSc(0)c(0)e -, n q J" q r q=l r=l 6 -I ) (33) The analogous output from the cross-correlation system is P 2 (l, 0, S x e e, . ji f * R R • • s o e R , D ^) = Re i A (e) B , (e) + s s s s R ' R £ q r q=l r=l £(e > - l(e„) A(9 ) B . (0 ) e q q f r R # -jl(9 ) r + A(0) 2 S B. (0 ) e r i r r=l t-j€ e i + B. (0) Z S A(0 ) e ^ , q q q=l (34) A comparison oJ » h<;se equations with those for a single interfering signal shows thai th< ten again has the low-level interference pattern van by Equation N9). The rejection of Interference Li 23 S . <1 for all q will depend primarily on this pattern which, as was shown q in Section 3.2, can be quite poor even when the product pattern itself is good. The step from R discrete interfering signals to a continuous back- ground of interference is quite straightforward, and the rejection of low- level interference in this case also depends on the low-level interference pattern. A major hindrance in the design of a satisfactory interference pattern is the interferometer factor cos (P i sin 0) due to the separation i of the two antennas Its oscillations between +1 and -1 cause high sidelobes, especially if bi0 ) has high sidelobes also. Indeed, it is quite easy to show that it is impossible to get a product pattern and a low-level interference pattern that both possess low sidelobes if one of the factor patterns does not have them. If I is quite large the cosine factor in the noise term cos[P I sin + a(e ) - P(9 )] s lace^ b( 0l )' Ho^To") -(3(o o )i from Equation (18) will oscillate rapidly between -1 and +1 as varies Hence, for a number of values of this noise term will be given by !a(0 ) b(0 )1 S _ i y. sHa(0 ) b(0 )| cos[a(0°) - P(0°)] L X and if b(0 ) has high sidelobes for these values of the noise term will be large. It can be reduced only by making ia(0 )| small, much smaller th an Ibce^ However, the low-level interference noise term is given by 24 cos (I + a(0 ) _ (3(0°)) cos (Pi sin - | * a(0°) - (3(0 )) s ; a(0 ) __ _i 1 + |b(0 )| L_-i_ _ .J. 1 cos[a(0°) -P(0°)] cos;a(0 ) -(3(0°)] As £ varies this noise component varies also. It can be shown that the max- imum value is at least S \ ibO.M - la(0 )| r ___^_ i i__i_ S - [b(0)| - |a(0 n )| ' cos[a(0°) - (3(0°)] [ 1 l To minimize this component we therefore must make ia(0 )\ srf (b(0 , ) ■ , and this is not what is required for the suppression of the other noise compon- ent. It follows that one or the other of the noise components can be minimized but not both. 3.4 Interference in the Case of More Tnan One Multiplication The case of the multiplication of N antenna patterns with R interfering signals is obviously an involved situation. For N even the product pattern is N/2 . jPl sin R j| V 1 °< S l *, °! - *> - " A 2q -l (9)e * = V \q-l (9 j ) q=l v J=l (35) e jPi 2q-l Sin 9 j -jPi .me r -jft -^ 2q .i Bin e k > A 2q (6) e ^IS k e A 2q (0 k ) e Bj way Oi b N i 4, R ■ 1, and let the patterns be real. Even with the expression that results when put into trigonometric 25 P (1, 0: Se , . i i i ) = A (0)A(0)A(9)A (6)cos<Pl sine-Pi sinG+Pl sine) 1 * "7, cos iPi sine -Pi sin0+Pi' sin0) 2 3 4 A (0 ) 2 1 /A (0 ) S ! -r- 7 -r— cos(Pi sin0-Pi sin© A ( e ) 2 3 1 1 + Pi 4 sin0 + l 1 ) + a~7q7- cos(Pi 2 sin0 1 - Pi 3 sin0 4- 0| sine - ^) A (0 ) t A / Q x ~ cos (Pi sin0 - Pi sin0 1 + Pi sine 4 ^) W \ r-7r T - cos (Pi sine - Pi sine + Pi sine 1 - (L) A (0) 2 3 41" 4 A (0 ) A (0 ) A (0) A 2 (0T- C ° S(Pi 2 5ine i " Pi 3 Sin6 + Pi 4 Sin9) J. u W A 3 (0 1 ) -nenrier cos((3i 2 sine - Pi 3 slne i + pi 4 sin9 - 2 V J- o VV A (6 )* a (0TA-T0T cos((3 V inG " Pi 3 sine + p V in V 1 4 W A 3 ,0 1 ) 1~(e) A (0) cos(Pi 2 sin 0i - Pi 3 sin0 1 + Pi^in©) ~K^Q) A 4 (0) cos (Pi ^^ - Pi 3 sin t Pi 4 sin ei f 2^) VV A 4 ' e i ) I ^ 3 T0TA 4 l0T~' COS(,3i 2 Sin0 " P V ine ! + Pl 4 Sine i , | 26 [A (0 ) A (0 ) A (0 ) \m 1(8) v(e> —tfV-i " Pi 3 sine i " p V ine - V A (G ) A (9 ) A (B ) V(8) 1(8) A*<9) c ° s(! V in9 - P V ln9 l - P V in6 l ' V A (0 ) A (8 ) A (0 ) \ (9) 1(9) A* (9) c -«V ine i - ? V in6 * P V ln9 l " V 12 4 A (9 ) A (6 ) A (8 ) \ C 9 ) A (6) A*(9) <"»«V 1 '* 1 " P's'^! + "V 1 " 8 ! 2 3 4 ,! , A (0 ) A (0 ) A (0 ) A (0 ) -4 I 1 1 2 1 3 1 4 1 ,n. . _ Tl) A o (0) A (9) A f8) " cos(Pi 2 Slne i 12 3 4 Pi sin0 + Pi sine, ) 4 1 I (36) 2 3 There are no* 3 distinct interference patterns associated with S, S and S 4 in addition to the product pattern associated with S . It follows a fortiori that care must be exercised in the design of product patterns involving severa] multiplicative operations if interference is to be rejected. It is important to note that even in the case where time averaging is 2 possible, there are still terms in the S interference pattern that are not functions of £i t ) and hence cannot be averaged out. The terms of the S and S patterns are all functions of £.(t) and can be removed. If 6 is and the patterns ire normal ized, the time-averaged output reduces to iyi, o° s, e t v i 3 , i> 4 ) = i + s 2 [■ 1 ce 1 ) a. 2 (e 1 ) coscPi^mo^ B) coB(Pl 4 «in6 1 ) • :i 2 U) \ ) '•/ ,J i ) cos (Pi si no - Pi sine J 27 * (6,) MM cos (-(3| sine, + Pi .sine.) 3141 31 41 + S 4 a ce n ) a (6. ) a O-.) a (0 ) cos (Pi sine - Pi sin0 * Pi sine. ) (37) in which we see that the S interference pattern is a sum of 4 cross products, e.g., a (0 ) a (0 ) cos(Pi sin - Pi sin ), of the original patterns ^ J. *J3 1 ^ X *j X modified as usual by a cosine term. A useful design would minimize this pattern in the sidelobe region as well as the product pattern itself. We note also that if N is odd the random phase angles from the signal as well as the noise are present in all output terms and consequently time averages of the product pattern systems for N odd all go to zero. Hence, only an even number of antenna patterns can be cross-correlated in say a Radio Astronomy application. 28 4. SUPPRESSION OF THE EFFECT OF AVERAGE BACKGROUND NOISE BY USE OF THE NEGATIVE SIDELOBES OF THE PRODUCT PATTERN An interesting and useful feature of product patterns is their negative sidelobes. Thus they contrast with the power patterns of additive antennas which are non-negative functions of x = Pi sin 8./2„ It will be shown that for an arbitrary distribution of background noise in the absence of the signal the pattern can be adjusted so as to reduce to zero the time-averaged response to this noise in the system output. We assume that the noise coming from any direction is a narrow band quasi-stationary random process, statistically independent of the noise coming from any other direction. If its average power density is given by S (x, t) (which is generally a slowly varying unkno\ function), then the time-averaged response of the additive antenna system to this noise is I (x , t) = o' S 2 (x, t)|C (x - x)! 2 dx (38) 7 o 2 where . C(x) " is the power pattern of the antenna with x corresponding to the o / (3£ s i n 9 \ 2 main lobe direction I x = — . S (x } t) is clearly non-negative and the integral will be positive for any noise background that is not identically zero for all x. For the produd pattern case, the time-averaged output is I (x , t) = J r S 2 (x, t) P o (x - x) dx (39) and since P, (x) can tak'.- on negative values it should be possible to select iny given time t the integral equation 29 r S 2 (x, t ) P (x - x) dx = (40) ' o 2 o To make Equation (40; hold for all t it is clear that P (x) must become a slowly varying function of time in order to compensate for the variation 2 in S (x, t). Thus Equation (40) can be generalized to r S 2 (x, t) P (x - x, t) dx = (40a) J 2 o In addition, as the beam angle scans it will be necessary, in general, for the pattern to change if the output due to the noise is to remain at zero. In this derivation it is assumed that it is known a priori when the signal is present and when it is absent. During the periods when it is absent the antenna pattern is adjusted to eliminate the effect of the slowly varying average background noise. Then when the signal is transmitted its effect alone will be observed in the system's output, A slowly varying background noise density will gradually change this ideal condition and so there should be some prearranged sequence of time intervals in which the signal is turned off and the receiver pattern modified to compensate for this gradual change of background noise density. To illustrate, let us consider the Compound Interferometer with the uni- formly weighted aperture replaced by a linear array of m * 1 elements with real and symmetric weighting (a = a where r indicates the r element to r -r' the right from the center of the array and -r indicates the r element to the lef t; : The patterns of the two antennas are m/2 A(x - x ) = 2_ a cos[2r(x - x )] (41) o r=0 r *• o Dl B(x - x ) = cos[ m(x - x )] o o 30 (42) R« ■ n Pi sin Pi sin 9 o where x = - — — ■ — — , x = ■ 2 ' o 2 and G is the beam angle of the patterns, o Now from Equation (3a) it follows that m/2 P (x - x ) = 2 a cosi r(x - x )] cos[m(x - x )] cos[2m(x - x )] 2 o r o o o r=0 (43) m/2 m/2 = 2af(x-x)=2 af(x-x) rr o rro r=0 r=0 Substituting this into the expression for the output due to the arbitrary background noise density (Equation (39)) we get I(x , t) = o' m 2 = 2 r=0 2 m/2 S (x, t) 2 a f (x - x) d: rro r=0 S (x, t) f (x - x) dx r o (44) and from Equation i40) we require that m 2 2 a , ' 2 m/2 S x t) f fx - x) dx = 2 a q (x , t) = r o r r o' r=0 (45) where q (x t ) r o S (x, t) f (x - x ) dx r o (46) To get the coil I , q (x , l) it is necessary to install switches in the r o trar. from the various array elements. If all switches but the the center element' I Lne were open then the output from the system i 31 I (x , t) = a q (x , t) (47) o o o o o and if during the measurement starting at, say, t , a is set equal to unity, we get q (x t ) = I (x t ) (47a) o <r o o o o Likewise opening; all but the lines from the a elements and setting a equal ft. r r to unity we get q (x , t ) = I (x , t ) (48) r o' o r o o In this way all of the coefficients can be obtained. We can think of the q (x , t)s as components of an — + 1 dimensional noise r o- 2 vector Q'x t 1 whose scalar product with the weighting vector A('x , t ) also o o o f _ a. i dimensions, is required to be zero by Equation (45) „ It is clear that there is no unique vector A(x , t ) that will satisfy this condition since o o any vector will do which lies in the hyperplane to which Q<x t ) is perpen- o o dicular Out of this infinity of vectors that satisfy Equation (45) it would be best to select one which gave the maximum response to the signal when it was incident at the beam angle 6 . This means that we pick Aix t ) such that o o o m/2 P (0) = 2 a (x , t ) = MAX (49) 2 r=0 However it is obvious that some limit on the size of the vectors A('x , t ) o o themselves must be established because if any vector is doubled in size then the sum of its components will also be doubled, Hence we seek out from all 32 vectors of a given length lying in the hyperplane the one whose sum of its components is the largest. We set this length (without loss of generality) equal to unity and we obtain the condition equation m/2 2 a (x , t ) = 1 (50) r=0 r ° ° Note that there are — + 1 coefficients a (x , t ) and still only 3 2 r o o simultaneous equations. There is thus a possible — - 2 additional conditions that could be satisfied, e.g. 1) first null occurs at 9 - A, 2) first sidelobe has maximum at = TT , etc. However, let us consider the five-element case, m = 4 in which there are just three possible conditions; the three equations are a (x , t ) + a, (x , t ) 4 a (x , t ) = MAX o o' o loo 2 o 7 o c (x , t ) a fx , t ) + c, (x , t ) a (x , t ) f c (x , t ) a <x , t ) - oooooo lo'oloo 2 o o 2 o o 2 2 2 a (x , t ) + a, (x , t)+a„(x.t)=l o o' o loo 2o'o (51) The last two equations represent a plane and a sphere, respectively, in three dimensional space. Their intersection is a circle of unit radius centered at the origin and a certain point (or points) on it satisfy the first equation. To get this point I arrange' s method of undetermined multipliers can be used and the following s< I -,\ equal ions results. L + c (x , t ) X (x , t ) + 2a (x , t ) X (x , t ) » -/ o o o O o o 33 1 .. c ix , t i X (x , t ) + 2a, (x . t ) \ (x t ) = loo ooo loo lo o (52) 1 - c (x , t ) v 'X , t ) + 2a (x , t ) X (x , t ) = 2 " o o ooo 2 o o loo c (x , t ) a <x , t ) + c, (x , t ) a n (x , t ) + c„ (x t ) a (x t ) = ooo o o o 1 o' o 1 o' o 2 o 7 o 2 o o 2 2 2 a ix t ) + a (x , t ) + a„ (x , t ) = 1 o o' o 1 o' o 2 o' o A digital computer could be used to solve these equations for the weighting coefficients a ( x s t ) a (x , t) and a (x , t ) Then instructions from ooolo'o 2 o o the computer could be fed back to each of the lines coming from the array elements where a variable attenuator would be adjusted to give the correct weighting. A diagram of the system is shown in Figure 5. Note that the computer also sends phase shift instructions (for scanning) and switching instructions i for obtaining the q (x , t ) coefficients) The conditions r o o fed into the computer are, of course, those that are mentioned above and which give rise to Equations (52) which the computer must solve. A final word of caution should be mentioned Although in theory the equations can always be solved to reduce the noise output to zero, some noise backgrounds can be eliminated only by resorting to a supergain condition For example, a not uncommon case is that of a single point source of noise which can be eliminated only if the pattern always has a null in the direction of the point source This presents no difficulty until the beam angle x itself approaches the angle of the point source. This requires the null to approach the beam angle, and since the pattern is an analytic function the response of the pattern to the signal incident at the beam angle will approach zero. V 3 r-H > CO bfl cd 0) > < CO x: +j <H O c o •H CO W a <x 3 CO (1) XI 0J in •H o c •o c 3 o bD o CQ <H o c o •H +J 3 X3 cd (-< O -H U o a) o -t-> en co >> A CO -P (D 3 be •H 3 >> a m •m cd 3£ 35 To maintain the response at, say, unit amplitude it will be necessary to increase the magnitude of the weighting coefficients a (x , t ) which means going to a supergain condition. Furthermore, if the beam angle and the angle of the point source of noise coincide it will be necessary to split the beam to satisfy Equation (45). This means that it is impossible to eliminate the noise and not the signal if both are incident as plane waves at the same angle. How- ever, there is usually more than one source of noise and, in theory at least, the signal from a point source which may or may not coincide with one of the noise sources can be retained and the noise rejected. 36 5. THE NOISE LIMITATION ON THE FORMATION OF ARBITRARY PATTERNS FROM A TWO ELEMENT INTERFEROMETER BY MEANS OF MULTIPLICATIVE OPERATIONS 5,6 Several writers •' have discussed the possibility of forming arbitrary antenna patterns by multiplicative operations on the cross-correlated output from a single pair of elements spaced a distance I apart^ i.e., from a simple interferometer „ Patterns of arbitrary sharpness of the main lobe could be synthesized and used to advantage in direction-finding applications. However^ no account was taken of noise in these schemes^ and it will be shown below that noise severely limits their usefulness. From Equation (3a) the pattern of the interferometer is P (6, i) = cos (Pi sin 8) (58) It is raised to various powers from say to M and a linear combination of these terms each weighted by a factor q is formed M k Q (x) = 2 q cos x k=0 (59) where x = Pi sin 9 But k cos X k-1 k 1 l k cos[ (k - 2r)x] -4 — „ k k 2 la (60) if k is even, and k COS X k-1 2 | ') r, cos[ (k - 2r) x] (60a) 37 if k is odd and where is the binomial coefficient. Hence M Q (x) = 2 w cos(kx) k=0 where the w s are linear combinations of the q.'s. By a proper choice of the k K q :'s any arbitrary pattern of a 2M + 1 element array (M even) can be simu- lated. If M is odd the array has 2M elements. The assumption is made that there is no noise. In particular, it is assumed that only a single plane wave ; the signal ; is incident on the antennas, In the following it will be shown that if the signal is received in an arbitrary background of noise there will generally be a shift in the apparent direction of the signal as "seen" by the basic pattern cos (Pi sin 0) and the process of forming a "better" pattern by multiplicative operations cannot remove this error. Let the signal of say unit average power be incident on the system from direction x = Pi sin 9 and let the noise have an average power density given 2 by S (x) . Then the time-averaged output from the low-pass filter of the correlation system is I(x) = o ;5( X;l - x) + S^( Xl ), cos(x - x ) dx oil cos(x - x) + o S (x, ) cos x dx 1 11 cos X S (x ) sin x dx sin x (61) where x is the beam angle of the interferometer pattern. We see that the 38 terms in the square brackets are the Fourier coef f icients, a and b , of the series expansion of S(x) given by where S(x) = Z a cos kx + 2 b sin k x k=o k k=i k cos(kx) cos(jx) dx = 6 sin(kx) sin(jx) dx = 6, , kj sin(kx) cos(jx) dx = , and 5 kj =l, k-J = 0, k £ j Therefore Equation (60) can be expanded as I < x ) = cos(x - x) + a cos x + b sin x o o 1 o i o = cosCx - x) + c cos(x - x ) o l o c where 2 2 . 2 . -1 c, = a, + b, , x = tan 1 1 1 c The arbitrary nod ie background is "seen" by the pattern as a point source at 39 ang le x = Pi sin with average power c . c c 1 Equation (63) can be reduced finally to I(x ) = [ cos x + c cos x 1 cos x + [ sin x + c sin x ■] sin x o 1 c J o 1 c o d, cos(x - x ) 1 o d (66) where 2 ,2 [cos x + c cos x + [sin x + c, sin x J L l c J L 1 c J x = tan d -1 sin x + c sin x 1 c_ cox x + c cos X 1 c. (67) The relations are best shown by the phasor diagram of Figure 6. The output is jx the scalar product of the unit scanning phasor e and the phasor sum e ' + J x ^ J x h c d c e = de There is a magnitude error and an angular error in the esti- mated value of the signal. However^ if x = x or x -77 the angular error is ze ro; the magnitude error is maximized and is - c. . Contrariwise if x = 1 c x - cos (c ,/2) the amplitude of d is unity and no magnitude error occurs. The maximum error in angle for a given c < 1 is (x -x) =- sin c 1 d max 1 and occurs at x = x + (77/2 + sin e, ) . If c ' • 1 the error can be as much c 11 — as 77, The basic patterns due to the signal alone ; the noise alone and the signal plus noise are shown in Figures 7a, 7b ? and 7c respectively. In Figure 7d a pattern Q„(x - x ) synthesized from the basic signal plus noise pattern. MOd or- r , d cos(x - x ) is shown. The error (x, - x ) occurring in the basic pattern 1 o d d can in no way be reduced by this synthesizing of Q (x - x ) since the process Mod 40 0) a SI. in o o c 3 Z M H ^< —i 0. in o < 3 x-a — 1 r-l (U CO CU c 05 a bO •^1 60 r-l i-( ■H ffl CO z CO C bo <M VI <H -H O CO c c C <H u u U O 0) 4) 0) 4-> •P P c ■p ■»-> P u ed 0) 3) CO 0, £ a, *> p k >-• t< in at 1 0) 0) a) a, •p ■u p 0) 0) d) T3 s G E a» ■s. o « V ^ l* U -H -^ 0> CO 0) in <H <*-{ <H 0) u ^ fc £ <D 0) 1) P +) P P c —^ a G c >> J w i—i HH CO d A u Q D O "O \ \\ tr M, UJ CO CO < 1 1 z I w 1 X *J ^1 a: \ 0" / / ^ _l z CO CO O CO < X 0. \ 1 \ \ \ AV \ \ \ / \ \ ^ ' / \ ^/ ^ ^ — — \ X \ \ z / \ "O \ 2 Z CO / V *— *• \ < <\/ \ X\ 1 7\. \ 1 \ X ^«- — ► to Q- / ' ^ Xy w\S\ \ \ \ \ "°v •-* G ct) V C B hfi 0) -H r-H w W CO O JG p ^ <H m O a c h >. ■H m c« *-> ^ fit *-> U P D < C D. CO ♦^ <x> 3 > t< *-> a <u Cfl <D Ul U IX •H H O r-< u ^. D. •H '/i U5 •(-> Tj g rH x: .-1 3 E a fl 'D Cb ^ 3, 41 only defines more sharply the value of x for the benefit of the observer. The synthesized pattern is limited not by its own shape and sharpness (in theory it can be made to approach a delta) but by the basic interferometer pattern to which any combination of point source plus background noise appears as another equivalent point source. This is all that the synthesized pattern will ever "'see". If there is a fairly low signal to noise ratio it can be concluded that a method of pattern syntheses such as this can lead to large errors in the estimated value of the angle of incidence of the signal . 42 6. CONCLUSIONS It has been shown that the cross-correlation of antenna voltages to obtain product patterns gives rise to two major problems: 1. Multiplicative distortion of amplitude modulation, 2. Interference patterns which differ from the original product pattern and which can have high sidelobes even when the product pattern itself has low sidelobes. The first problem can in theory be overcome by a rather complicated system which requires frequency shifting and filtering. An analysis of the second problem shows that if instantaneous reception of, say, modulated signals is required, special care must be exercised in the design of both product and interference patterns if an adequate rejection of interference is to be achieved. For systems in which the time average of the incoming signals is all that is required the problem simplifies considerably but if N, the number of patterns to be multiplied, is large it is still quite involved. It has also been shown that product patterns possess a distinct advantage over conventional additive power patterns in that their negative sidelobes can be used to suppress the integrated effect of any arbitrary distribution of background noise. Finally, the artificial formation of arbitrarily sharp antenna patterns by multiplicative operations on the time-averaged output of a two element interferometer system has been shown to be of limited value in the presence of ,i background noise distribution because the noise causes a shift in the apparent direction of arrival of the signal. This shift is not reduced by formation '<( i more directive" pattern whose effect is therefore nothing 43 more than to emphasize the apparent direction of arrival of the signal as indicated by the interferometer pattern. If the noise is relatively strong the error in the apparent direction of arrival can be quite large. 44 REFERENCES 1. Mills, B. Y. and Little, A. G., "A High-Resolution Aerial System of a New Type." Austral. J. Phys. , Vol. 6, pp. 272-278, September, 1953. 2. Covington, A. E. and Broten, N. W "An Interferometer for Radio Astronomy with a Single Lobed Radiation Pattern," IRE Trans, PGAP, AP-5, No. 3, pp. 247-255, July, 1957. 3. Welsby, V, G. and Tucker, D. G. , "Multiplicative Receiving Arrays," J, Brit, IRE, 19, pp. 369-382, June, 1959 4. Berman, A. and Clay, C. S., "Theory of Time-Averaged Product Arrays," J. Acoustical Soc. America , 29, No. 7, pp. 805-812, August, 1957. 5. Drane, C. J,, Jr., "Phase Modulated Antennas," presented at IRE-URSI Joint Fall Meeting, 1959, San Diego, California. 6. Brown, J. L M Jr. and Rowlands, R. 0., "Design of Directional Arrays,' J. Acoustical Soc. America , 31, No. 12, pp. 1638-1643, December, 1959. 7. Dwight, H. B., Tables of Integrals and Other Mathematical Data , The Macmillan Company, p.l. ANTENNA LABORATORY TECHNICAL REPORTS AND MEMORANDA ISSUED "Synthesis of Aperture Antennas, " Technical Report No : 1, C.T.A, Johnk, October. 1954. * A Synthesis Method for Broad-band Antenna Impedance Matching Networks," Technical Report No. 2, Nicholas Yaru, 1 February 1955 The Asymmetrically Excited Spherical Antenna," Technical Report No, 3, Robert C Hansen. 30 April 1955,* "Analysis of an Airborne Homing System," T ech nical Report No, 4, Paul E„ Mayes r 1 June 1955 ( CONFIDENTIAL) . Coupling of Antenna Elements to a Circular Surface Waveguide, Technical Report No 5, H. E King and R. H. DuHamel , 30 June 1955.* Axial ly Excited Surface Wave Antennas," Technical Report No. 7, D, E. Royal, 10 October 1955 * Homing Antennas for the F-86F Aircraft f 450-2500mc) , " Technical Report No. 8, P.E t Mayes R.F, Hyneman, and R.C. Becker, 20 February 1957, (CONFIDENTIAL). Ground Screen Pattern Range," Technical Memorandum No 1, Roger R. Trapp, 10 July 1955 * ~~~ ~~~ Conj rag t AF33 ( 61 6 i ^3220 Effective Permeability of Spheroidal Shells," Technical Report No. 9, E. J. Scott and R H DuHamel, 16 April 1956. An Analytical Study of Spaced Loop ADF Antenna Systems," Technical Report No. 10, D G Berry and J B, Kreer, 10 May 1956. "A Technique for Controlling the Radiation from Dielectric Rod Waveguides," Tech_nical_Repot No. 11, J W. Duncan and R = H DuHamel, 15 July 1956.* Directional Characteristics of a U-Shaped Slot Antenna, ' Technical Report No^_12, Richard C Becker, 30 September 1956 „** Impedance of Ferrite Loop Antennas," Technical Report No 13, V, H, Rumsey and W. L Weeks. 15 October 1956 "Closely Spaced Transverse Slots in Rectangular Waveguide," Technical Report No^_l_4, Richard F Hyneman, 20 December 1956. Distributed Coupling to Surface Wave Antennas," Technical Report No 15. jj^ ' — — — * Ralph Richard Hodges, Jr , 5 January 1957. The Characteristic Impedance of the Fin Antenna of Infinite Length, Technical Report No 16 Robert 1 Carrel, 15 January 1957 On the Estimation of Ferrite Loop Antenna Impedance, Technical Re port No. 17, Walter L. Weeks, 10 April 1957.* A. Note Concerning a Mechanical Scanning System for a Flush Mounted Line Source Antenna fechnic .1 Report No. 18, Walter L. Weeks, 20 April 1957, Broadband logarithmically Periodic Antenna Structures. Technical Report No . 19 R K DuHamel and D E. Isbell, 1 May 1957. Frequency Independent Antennas," Tec hnical Report No 20 V. H Rumsey, 25 October 1957 The Equiangular Spiral Antenna," Techni cal Report No, 21, J D, Dyson, 15 September 195? Experimental Investigation of the Conical Spiral Antenna Technical Report Vo 22. R L. Carrel 25 May 1957.** ' """ Coupling between a Parallel Plate Waveguide and a Surface Waveguide," Technical Report \ T o. 23 E ' Scott, 10 August 1957. Launching Efficiency of Wires and Slots for a Dielectric Rod Waveguide," rechni .. | i 24 J. W. Duncan and R. H DuHamel, August 1957. The Characteristic Impedance of an Infinite Biconical Antenna of Arbitrary Cross Section, hnical Repo rt No. 25, Robert L Carrel, August 1957. -Backed Slot Antennas, ' Technical R eport No. 26_, R. J. Tector, 30 October 1957 Coupled Waveguide Excitation of Traveling Wave Slot Antennas/ Techni cal Re] [0 27 W I Weeks, 1 December 1957. Pti Red ingular Waveguide Partially Filled with Dielectric," 28 W I Weeks, 20 December 1957 p< r lint Length of Biconical Structures of Arbitrary Cross rechnli i] Rep ort No. 29 , J. D Dyson. 10 January 1958 Lcally Periodic Antenna Structur Technical Report No. 30, D ^58 ( ' pular Slots,' technical Reporl ^'< 31, N. J. ; , L958 tation oi i Surface Wave on a Dielectric Cylinder/' I w Du J5 May L058 "A Unidirectional Equiangular Spiral Antenna, " Technical Report No. 33, J. D. Dyson, 10 July 1958 Dielectric Coated Spheroidal Radiators," Technical Report No. 34^ W. L Weeks, 12 September 1958. "A Theoretical Study of the Equiangular Spiral Antenna," Technical Report No. 35, P E Mast, 12 September 1958. Contract AF33 '616) -6079 Use of Coupled Waveguides in a Traveling Wave Scanning Antenna,' Technical ILERELLJ^JL 6 -.' R H MacPhie, 30 April 1959. On the Solution of a Class of Wiener-Hopf Integral Equations in Finite and Infinite Ranges, Technical Report No. 37, Raj Mittra, 15 May 1959. "Prolate Spheroidal Wave Functions for Electromagnetic Theory," Technical Repojr£_No ^_38 , W, L, Weeks, 5 June 1959. Log Periodic Dipole Arrays," Technical Report No. 39, D E Isbell, 1 June 1959. A Study of the Coma-Corrected Zoned Mirror by Diffraction Theory," Technical Report No^ 40. 3. Dasgupta and Y. T„ Lo, 17 July 1959 The Radiation Pattern of a Dipole on a Finite Dielectric Sheet/" T echnical ReportNq 41 KG Balmain, 1 August 1959 The Finite Range Wiener-Hopf Integral Equation and a Boundary Value Problem in a Waveguide. Technical Report No. 42, Raj Mittra, 1 October 1959. "impedance Properties of Complementary Mul titerminal Planar Structures " Te chjrica.1 JRepor t No __43_, G, A, Deschamps, 11 November 1959 On the Synthesis of Strip Sources," Technical Report No 44 Raj Mittra, 4 December 1959 Numerical Analysis of the Eigenvalue Problem of Waves in Cylindrical Waveguides," Technical Report No^ 45. C H Tang and Y. T, Lo, 11 March I960, Ve* Circularly Polarized Frequency Independent Antennas With Conical Beam or Omnidirectional Patterns," Tec hnica l Report No, 46, J.D Dyson and P.E. Mayes, 20 June 1960 Logarithmically Periodic Resonant-V Arrays,'" Technical Report No, 47, P.E. Waves and R L Carrel, 15 July 1960. * Copies available for a three week loan period Copies no longer available. AF 33(616) -6079 DISTRIBUTION LIST One copy each unless otherwise indicated Commander Wright Air Development Center Attn: WCOSI, Library Wright-Patterson Air Force Base, Ohio Commander U.S. Naval Air Test Center Attn: ET-315, Antenna Section Patuxent River, Maryland Chief Bureau Naval Weapons Department of the Navy Attn: (RR-13) Washington 25, D.C. Commander USA White Sands Signal Agency White Sands Proving Command Attn: SIGWS-FC-02 White Sands, New Mexico Director Air University Library Attn: AUL-8489 Maxwell AFB, Alabama Army Rocket and Guided Missile Agency U.S. Army Ordnance Missile Agency Attn: ORDXR-OMR Redstone Arsenal, Alabama Commander Air Force Missile Test Center Attn: Technical Library Patrick Air Force Base, Florida Commander Aero Space Technical Intelligence Center Attn: AFCIN-4c3b, Mr. Lee Roy Hay Wright-Patterson AFB, Ohio Director Ballistics Research Lab. 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Dr A, Alford M F Contract AF33( 600)-36l08 299 Atlantic Avenue Boston 10, Massachusetts ATA Corporation 1200 Duke Street Alexandria, Virginia Bell Telephone Labs , Inc Attn: R, L. Mattingly M F Contract AF33(, 616) -5499 Whippany, New Jersey Bendix Radio Division Bendix Aviation Corporation Attn, Dr, K. F, Umpleby M [• (Onttact AF33( 600) -35407 Towson 4, Maryland Boeing Airplane Company Attn C , Arms! rong M/F Contract AF33( 600)-36319 7755 Marginal Way Seattle, Washington Doc i ng Ai rpl am C> irnpany Attn Robert Shannon W I Contrad AF33(600)-35992 Wichita, Ka nsas AF 33(616)-6079 Canoga Corporation M/F Contract AF08( 603) -4327 5955 Sepulveda Boulevard P.O. Box 550 Van Nuys, California Dome & Margolin, Inc. M/F Contract AF33(600)-35992 30 Sylvester Street Westbury Long Island, New York Dr. C. H. Papas Department of Electrical Engineering California Institute of Technology Pasadena, California Chance-Vought Aircraft Division United Aircraft Corporation Attn: R.C. 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I Mr Dwighl I Bbe] i 4620 Sunnyside Seal t : <■ 3, WashJ rig I on ITT Laboratories Attn;, L„ DeRosa M F Contract AF33( 616)-5120 500 Washington Avenue Nutley 10, New Jersey ITT Laboratories A Div. of Int, Tel. & Tel, Corp, Attn: G B S. Giffin, ECM Lab, 3700 E. Pontiac Street Fort Wayne, Indiana Jansky and Bailey, Inc Engineering Building Attn: Mr. D. C, Ports 1339 Wisconsin Avenue, N.W. Washington, D.C Jasik Laboratories, Inc, 100 Shames Drive Westbury, New York John Hopkins University Radiation Laboratory Attm Librarian M/F Contract AF33(616)-68 1315 St. Paul Street Baltimore 2, Maryland Applied Physics Laboratory Johns Hopkins University 8621 Georgia Avenue Silver Spring, Maryland Lincoln Laboratories Attn.. Document Room M/F Contract AF19(122)-458 Massachusetts Institute of Technology P.O. Box 73 Lexington 73, Massachusetts AF 33(616)-6079 Litton Industries, Inc. Maryland Division Attn Engineering Library M F Contract AF33( 600)-37292 4900 Calvert Road College Park, Maryland University of Michigan Aeronautical Research Center Attn: Dr, K, Seigel M/F Contract AF30( 602)-1853 Willow Run Airport Ypsilanti, Michigan Lockheed Aircraft Corporation Attn: C, D, Johnson M/F Contract NOa(s) 55-172 P.O. Box 55 Burbank, California Microwave Radiation Co,, Inc, Attn: Dr, M, J. Ehrlich M F Contract AF33(616'>-6528 19223 S. Hamilton Street Gardena, California Lockheed Missiles & Space Division (Attn: E, A, Blasi M/F Contract AF33( 600) -28692 & AF33(616)-6022 Department 58-15 Plant 1, Building 130 Sunnyvale, California The Martin Company Attn; W, A, Kee, Chief Librarian M/F Contract AF33(600)-37705 Library & Document Section Baltimore 3, Maryland Ennis Kuhlman McDonnell Aircraft P.O. Box 516 Lambert Municipal Airport St, Louis 21, Missouri Melpar, Inc . Attn, Technical Library M/F Contract AF19(604)-4988 Antenna Laboratory 3000 Arlington Blvd. Falls Church, Virginia Melville Laboratories Walt Whitman Road Melville, Long Island, New York Motorola, Inc , Attn: R, C, Huntington 8201 E, McDowell Road Phoenix, Arizona Physical Science Lab. Attn. R. Dressel New Mexico College of A and MA State College, New Mexico North American Aviation, Inc , Attn: J. D Leonard, Eng . Dept . M/F Contract NOa(s) 54-323 4300 E, Fifth Avenue Columbus, Ohio North American Aviation, Inc, Attn: H, A. Storms M/F Contract AF33( 600)- 36599 Department 56 International Airport Los Angeles 45, California Northrop Aircraft, Inc. Attn: Northrop Library, Dept. 2135 M F Contract AF33 (600)-27679 Hawthorne, California Dr, R. E, Beam Microwave Laboratory Northwestern University Evanston, Illinois AF 33' 616'. -6079 Ohio State University Research Foundation Attn: Dr, T, C, Tice M F Contract AF33(616)-6211 1314 Kinnear Road Columbus 8, Ohio University of Oklahoma Res, Inst. 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Attn technical Information Services 8433 Fallbrook Avenue P 0. Box 1006 Canoga Park California Rantec Corporation Attn: R. Krausz M/F Contract AF19( 604)-3467 Calabasas, California Raytheon Manufacturing Corp „ Attn; Dr, R, Borts M/F Contract AF33(604)-15634 Wayland, Massachusetts Republic Aviation Corporation Attn: Engineering Library M/F Contract AF33( 600)-34752 Farmingdale Long Island, New York Republic Aviation Corporation Guided Missiles Division Attn: J. Shea M/F Contract AF33(616)-5925 223 Jericho Turnpike Mineola, Long Island, New York Sanders Associates, Inc. 95 Canal Street Attn: Technical Library Nashua, New Hampshire Smyth Research Associates Attn.; J. B. Smyth 3555 Aero Court San Diego 11, California Space Technology Labs, Inc, Attn; Dr., R. C Hansen P.O. Box 95001 Los Angeles 45, California M/F Contract AF04( 647 ) -361 AF 33(616^-6079 Sperrv Gyroscope Company Attn B, Berkowitz M. F Contract AF33( 600)-28107 Great Neck Long Island, New York Stanford Electronics Laboratory Attn. Applied Electronics Lab. Document Library Stanford Iniversity Stanford, California Stanford Research Institute Attn; Mary Lou Fields, Acquisitions Documents Center Menlo Park, California Stanford Research Institute Aircraft Radiation Systems Lab, Attn D, Scheuch M F Contract AF33(616)-5584 Menlo Park, California Sylvania Electric Products, Inc. Electronic Defense Laboratory M/F Contract DA 36-039-SC-75012 P.0, Box 205 Mountain View, California Mr, Roger Battie Supervisor, Technical Liaison Sylvania Electric Products, Inc, Electronic Systems Division P.O. Box 188 Mountain View, California Sylvania Electric Products, Inc , Electric Systems Division Attn- C. Faflick M/F Contract AF33(038)-21250 100 First Street Waltham 54, Massachusetts Technical Research Group M/F Contract AF33< 61 6 ) -6093 2 Aerial Way Syosset, New York Temco Aircraft Corporation Attn: G, Cramer M/F Contract AF33( 600) -36145 Garland, Texas Electrical Engineering Res, Lab, University of Texas Box 8026, University Station Austin, Texas A, S 5 Thomas, Inc„ M/F Contract AF04C 645;-30 161 Devonshire Street Boston 10, Massachusetts Westinghouse Electric Corporation Air Arm Division Attn: P, D, Newhcuser Development Engineering M/F Contract AF33(600)-27852 Friendship Airport Baltimore, Maryland Professor Morris Kline Institute of Mathematical Sciences New York University 25 Waverly Place New York 3, New York Dr , S. Dasgupta Government Engineering College Jabalpur, M.P, India Dr , Richard C. Becker 10829 Berkshire Westchester, Illinois Tamar Electronics, Inc, Attn LB McMurren 2045 W Rosecrans Avenue Gardena, California The Engineering Library Princeton University Princeton, New Jersey AF 33(616)-6079 Dr. B, Chatterjee Communication Engineering Dept , Indian Institute of Technology Kharagpur (S.E. Rly,) India Sperry Phoenix Company Attn: Technical Librarian P.O. Box 2529 21111 North 19th Avenue Phoenix, Arizonia Dr. Harry Letaw, Jr„ Raytheon Company Surface Radar and Navigation Operations State Road West Wayland, Massachusetts